If that were true, the US would still have segregated lunch counters, grocery stores, and private buses. The Supreme Court may be getting us on the way there one day, but right now the only way that private businesses are allowed to discriminate against protected classes is to call the output work a "creative expression" like website design, floral arrangement, or cake decoration, and that's from the 303 Creative case.

Besides, how would it make sense if a company could bar you as a customer for being gay, but be compelled to employ you?